Integration of the intertwining operator for $h$-harmonic polynomials associated to reflection groups

Abstract:

Let $V$ be the intertwining operator with respect to the reflection invariant measure $h_{\alpha }^{2} d\omega$ on the unit sphere $S^{d-1}$ in Dunkl’s theory on spherical $h$-harmonics associated with reflection groups. Although a closed form of $V$ is unknown in general, we prove that \begin{equation*} \int _{S^{d-1}} Vf(\mathbf {y}) h_{\alpha }^{2}(\mathbf {y}) d\omega = A_{\alpha }\int _{B^{d}} f(\mathbf {x})(1-|\mathbf {x}|^{2})^{|\alpha |_{1} -1} d\mathbf {x}, \end{equation*} where $B^{d}$ is the unit ball of $\mathbb {R}^{d}$ and $A_{\alpha }$ is a constant. The result is used to show that the expansion of a continuous function as Fourier series in $h$-harmonics with respect to $h_{\alpha }^{2} d\omega$ is uniformly Cesáro $(C, \delta )$ summable on the sphere if $\delta > |\alpha |_{1} + (d-2)/2$, provided that the intertwining operator is positive.